On the matrix equation

Abstract

In this paper we investigate the matrix equation . Some sufficient and necessary conditions for the existence of Hermitian positive definite solutions as well as for the existence of the smallest Hermitian positive definite solution of the considered equation are derived. Two iterative methods for computing the smallest positive definite solution are proposed. Some numerical examples are presented to illustrate the theoretical results.

Keywords:

• Nonlinear matrix equation
• positive definite solution
• iterative method

AMS Subject Classifications:

• 15A24
• 65H05

Acknowledgements

I am grateful to Fernando De Terán and the anonymous referee for providing very useful suggestions, as well as to Prof. I. Ivanov for beneficial discussion and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.